13: Vector Functions
- Page ID
- 960
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A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector.
- 13.1: Space Curves
- We have already seen that a convenient way to describe a line in three dimensions is to provide a vector that "points to'' every point on the line as a parameter t varies. Except that this gives a particularly simple geometric object, there is nothing special about the individual functions of t that make up the coordinates of this vector---any vector with a parameter will describe some curve in three dimensions as t varies through all possible values.
- 13.2: Calculus with Vector Functions
- What makes vector functions more complicated than the functions y=f(x)y=f(x) that we studied in the first part of this book is of course that the "output'' values are now three-dimensional vectors instead of simply numbers. It is natural to wonder if there is a corresponding notion of derivative for vector functions.
- 13.3: Arc length and Curvature
- Sometimes it is useful to compute the length of a curve in space; for example, if the curve represents the path of a moving object, the length of the curve between two points may be the distance traveled by the object between two times.